To find a numerical value for the perimeter \( 6 + 4\sqrt{2} \), we'll approximate \( \sqrt{2} \):
\[
\sqrt{2} \approx 1.414
\]
Now, substituting this value into the expression for the perimeter:
\[
4\sqrt{2} \approx 4 \times 1.414 \approx 5.656
\]
Now, adding this to 6:
\[
6 + 4\sqrt{2} \approx 6 + 5.656 = 11.656
\]
It seems that this value doesn't match any of the provided options. Let's reassess the calculation while ensuring that we used the correct lengths.
From our earlier calculations:
- \( AB = 3 \)
- \( AD = 2\sqrt{2} \approx 2 \times 1.414 \approx 2.828 \)
Thus:
\[
\text{Perimeter} \approx 2 \times (3 + 2.828) = 2 \times 5.828 = 11.656
\]
We have confirmed our previous steps, but let's double-check the lengths of the sides.
Checking Distances Again:
-
AB: \( \sqrt{(2-2)^2 + (3-0)^2} = \sqrt{0 + 9} = 3 \)
-
AD: \( \sqrt{(4-2)^2 + (2-0)^2} = \sqrt{(2)^2 + (2)^2} = \sqrt{8} = 2\sqrt{2} \approx 2.828 \)
-
BC (parallel to AD): Length also equals approximately 2.828.
-
CD (parallel to AB): Length also equals 3.
Verifying the perimeter again:
\[
\text{Perimeter} \approx 2 \times (3 + 2.828) \approx 2 \times 5.828 \approx 11.656
\]
Since the options provided are significantly lower than our derived perimeter, we may conclude that the given vertices do not form a parallelogram, as the calculations clearly point towards the sides creating an incorrect perimeter based on the options. Thus, it's likely some data is inconsistent.
However, if we consider that perimeter must match one of the given options, I must double-check the orientation of vertices.
Let's summarize:
Given those lengths:
- AB = 3
- AD (or BC) = \( 2\sqrt{2} \approx 2.828 \)
- So, the preliminary calculation stands valid.
If the answer options are limited to the provided choices, the closest one doesn't directly relate, as they all fall below expected values based on traditional geometric input.
For further alignment with the given choices, a check on whether to take average or adjusted less than typical occurrences leading to lower gained perimeter might clarify results—should these be thought of differently or applied differently given numbers beyond standard constructs.
Conclusion:
Based on the correct formulations, there appears a significant deviation from given choices, always yielding a considerable difference, specifically a potential misalignment with provided vertex inset details faced by teaching or classroom subject matters producing outcomes refined bureaucratically.
If you have confidence in the coordinates given or they were read versus interpreted—there lies further said suggestion for insight!
